On the Equivalence of Cubic Permutation Polynomial and ARP Interleavers for Turbo Codes

Abstract

Recently, it was shown that the dithered relative prime interleavers and quadratic permutation polynomial (QPP) interleavers can be expressed in terms of almost regular permutation (ARP) interleavers. In this paper, the conditions for a QPP interleaver to be equivalent to an ARP interleaver are extended for cubic permutation polynomial (CPP) interleavers. It is shown that the CPP interleavers are always equivalent to an ARP interleaver with disorder degree greater than one and smaller than the interleaver length, when the prime factorization of the interleaver length contains at least one prime number to a power higher than one and it fulfills the conditions for which there are true CPPs for the considered length. When the prime factorization of the interleaver length contains only prime numbers to the power of one, with at least two prime numbers $p_{i}$ , fulfilling the conditions $p_{i}>3$ and $3 \nmid (p_{i}-1)$ , values of disorder degree smaller than the interleaver length are possible under some conditions on the coefficients of the second and third degree terms of the CPP.